VHEeP: A very high energy electron-proton collider

Based on current CERN infrastructure, an electron--proton collider is proposed at a centre-of-mass energy of about 9 TeV. A 7 TeV LHC bunch is used as the proton driver to create a plasma wakefield which then accelerates electrons to 3\,TeV, these then colliding with the other 7 TeV LHC proton beam. Although of very high energy, the collider has a modest projected integrated luminosity of $10-100$ pb$^{-1}$. For such a collider, with a centre-of-mass energy 30 times greater than HERA, parton momentum fractions, $x$, down to about $10^{-8}$ are accessible for photon virtualities, $Q^2$, of 1 GeV$^2$. The energy dependence of hadronic cross sections at high energies, such as the the total photon--proton cross section, which has synergy with cosmic-ray physics, can be measured and QCD and the structure of matter better understood in a region where the effects are completely unknown. Searches at high $Q^2$ for physics beyond the Standard Model will be possible, in particular the significantly increased sensitivity to the production of leptoquarks. These and other physics highlights of a very high energy electron--proton collider are outlined.